A class of De Giorgi type and local boundedness

Duchao Liu, Jinghua Yao


Under appropriate assumptions on the $N(\Omega)$-function, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev spaces. As the applications, the local boundedness property of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the local boundedness of the weak solutions for a class of fully nonlinear elliptic equations is provided.


Musielak-Sobolev space; local bounded property

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E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: The case 1 < p < 2, J. Math. Anal. Appl. 140 (1989), 115–135.

T. Adamowicz and O. Toivanen, Hölder continuity of quasiminimizers with nonstandard growth, Nonlinear Anal. 125 (2015), 433–456.

R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

A. Benkirane and M. Sidi El Vally, An existence result for nonlinear elliptic equations in Musielak–Orlicz–Sobolev spaces, Bull. Belg. Math. Soc. 20 (2013), 1–187.

S. Byun, J. Ok and L. Wang, W 1,p( · ) -regularity for elliptic eqautions with measurable coefficients in nonsmooth domains, Comm. Math. Phys. 329 (2014), 937–958.

F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator, Nonlinear Anal. 74 (2011), 1841–1852.

L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Func. Anal. 256 (2009), 1731–1768.

L. Diening, B. Stroffolini and A. Verde, Everywhere regularity of functionals with ϕ-growth, Manuscripta Math. 129 (2009), 449–481.

L. Diening, B. Stroffolini and A. Verde, Lipschitz regularity for some asymptotically convex problems, ESAIM Control Optim. Calc. Var. 17 (2011), 178–189.

T.K. Donaldson and N.S. Trudinger, Orlicz–Sobolev spaces and imbedding theorems, J. Func. Anal. 8 (1971), 52–75.

X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008), 1395–1412.

X.L. Fan, Differential equations of divergence form in Musielak–Sobolev spaces and subsupersolution method, J. Math. Anal. Appl. 386 (2012), 593–604.

X.L. Fan, An imbedding theorem for Musielak–Sobolev spaces, Nonlinear Anal. 75 (2012), 1959–1971.

X. Fan and C. Guan, Uniform convexity of Musielak–Orlicz–Sobolev spaces and applications, Nonlinear Anal. 73 (2010), 163–175.

X. Fan and D. Zhao, A class of de giorgi type and Hölder continuous, Nonlinear Anal. 36 (1999), 295–318.

X.L. Fan and D. Zhao, On the generalized Orlicz–Sobolev space W k,p(x) (Ω), J. Gansu Educ. College 12 (1998), 1–6.

M. Garcı́a-Huidobro, V.K. Le, R. Manásevich and K. Schmitt, On principle eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, Nonlinear Differential Equations Appl. 6 (1999), 207–225.

P. Harjulehto, P. Hästö and O. Toivanen, Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var. 56 (2017), no. 22.

D. Liu, B. Wang and P. Zhao, On the trace regularity results of Musielak–Orlicz–Sbolev spaces in a bounded domain, Comm. Pure Appl. Anal. 15 (2016), 1643–1659.

D. Liu and P. Zhao, Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces, Nonlinear Anal. Real World Appl. 26 (2016), 315–329.

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer, Berlin, 1983.


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